elaborate n(aub) = n(a) + n(b) - n(anb) with diagram

The equation \( n(A \cup B) = n(A) + n(B) - n(A \cap B) \) is a formula used in set theory to find the number of elements in the union of two sets, \( A \) and \( B \). Let's break down each part of this equation and explain it with the help of a Venn diagram.

### Explanation of Terms:

1. **\( A \cup B \)**: The union of sets \( A \) and \( B \) includes all elements that are in \( A \), \( B \), or both. We want to know the total count of unique elements in this union.
  
2. **\( n(A) \)**: This represents the number of elements in set \( A \).

3. **\( n(B) \)**: This represents the number of elements in set \( B \).

4. **\( A \cap B \)**: The intersection of sets \( A \) and \( B \) includes only those elements that are present in both sets \( A \) and \( B \).

5. **\( n(A \cap B) \)**: This represents the number of elements in the intersection of \( A \) and \( B \), or the count of elements shared by both sets.

### Why We Subtract \( n(A \cap B) \):

When we add \( n(A) \) and \( n(B) \) together, we count each element in \( A \cap B \) twice — once as part of \( A \) and once as part of \( B \). To avoid this double-counting, we subtract \( n(A \cap B) \), so each element is counted only once in the total.

### Visual Explanation with Venn Diagram

Imagine a Venn diagram with two overlapping circles:

1. **Circle \( A \)** represents set \( A \).
2. **Circle \( B \)** represents set \( B \).
3. The overlapping area of the two circles represents \( A \cap B \) — elements that are in both \( A \) and \( B \).
4. The area covered by both circles represents \( A \cup B \) — all elements in either \( A \) or \( B \), including their intersection.

Here’s how the formula works visually:
- **\( n(A) \)**: Count of all elements within Circle \( A \), including those in the overlap.
- **\( n(B) \)**: Count of all elements within Circle \( B \), including those in the overlap.
- **Subtracting \( n(A \cap B) \)** removes the overlap area, ensuring we don't count those elements twice.

### Example

Suppose:
- \( n(A) = 5 \)
- \( n(B) = 6 \)
- \( n(A \cap B) = 2 \)

Using the formula:
\[
n(A \cup B) = 5 + 6 - 2 = 9
\]

### Summary

The formula \( n(A \cup B) = n(A) + n(B) - n(A \cap B) \) ensures that elements in the overlap are only counted once in the total count of elements in the union.

Would you like to see further details or have any questions?

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### Related Questions

1. How can this formula be extended to three sets?
2. What is the significance of the intersection in probability?
3. How does the inclusion-exclusion principle apply in set theory?
4. How can we use this formula in real-world applications, like survey data?
5. What happens when two sets have no intersection?

### Tip
For three or more sets, the inclusion-exclusion principle helps adjust for overlapping elements across multiple sets, accounting for all shared elements without double-counting.

Explore the formula n(A ∪ B) = n(A) + n(B) - n(A ∩ B) in set theory, a key concept in understanding union and intersection of sets. Includes a detailed explanation and a Venn diagram illustration for clear visualization, suitable for students in Grades 9-12.

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